Related papers: Generalized Bessel and Frame Measures

The main result of this note, Theorem 2, is the following: a Borel measure on the space of infinite Hermitian matrices, that is invariant under the action of the infinite unitary group and that admits well-defined projections onto the…

We consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces $X$ homeomorphic to $\mathbb R^2$. Given a measure $\mu$ on such a space, we introduce $\mu$-quasiconformal maps $f:X \to \mathbb…

Let $(\Omega,\mathcal{F})$ be a standard Borel space and $\mathcal{P}(\mathcal{F})$ the collection of all probability measures on $\mathcal{F}$. Let $E\subset\Omega\times\Omega$ be a measurable equivalence relation, that is,…

Consider a mixing dynamical systems $([0,1], T, \mu)$, for instance a piecewise expanding interval map with a Gibbs measure $\mu$. Given a non-summable sequence $(m_k)$ of non-negative numbers, one may define $r_k (x)$ such that $\mu (B(x,…

We study the set S of ergodic probability Borel measures on stationary non-simple Bratteli diagrams which are invariant with respect to the tail equivalence relation. Equivalently, the set S is formed by ergodic probability measures…

We investigate in this work some situations where it is possible to estimate or determine the upper and the lower $q$-generalized fractal dimensions $D^{\pm}_{\mu}(q)$, $q\in\mathbb{R}$, of invariant measures associated with continuous…

We consider elliptic equations of the form (E) $-Au=f(x,u)+\mu$, where $A$ is a negative definite self-adjoint Dirichlet operator, $f$ is a function which is continuous and nonincreasing with respect to $u$ and $\mu$ is a Borel measure of…

Given two functions $f,g:I\to\mathbf{R}$ and a probability measure $\mu$ on the Borel subsets of $[0,1]$, the two-variable mean $M_{f,g;\mu}:I^2\to I$ is defined by $$ M_{f,g;\mu}(x,y) :=\bigg(\frac{f}{g}\bigg)^{-1}\left( \frac{\int_0^1…

The regular open subsets of a topological space form a Boolean algebra, where the `join' of two regular open sets is the interior of the closure of their union. A `credence' is a finitely additive probability measure on this Boolean…

The aim of this paper is twofold. Firstly, we chatacterize the Besov spaces $\dot{B}_{p,q}(\mathbb{R}^{n},\{t_{k}\})$ and the Triebel-Lizorkin spaces $\dot{F}_{p,q}(\mathbb{R}^{n},\{t_{k}\})$ for $q=\infty $. Secondly, under some suitable…

In this paper, we extend the concept of finite entropy measures in K\"ahler geometry. We define the finite $p$-entropy related to $\omega$-plurisubharmonic functions and demonstrate their inclusion in an appropriate energy class. Our study…

For each integer $n\ge 1$, denote by $T_{n}$ the map $x\mapsto nx\mod 1$ from the circle group $\mathbb{T} = \mathbb{R}/\mathbb{Z}$ into itself. Let $p,q\ge 2$ be two multiplicatively independent integers. Using Baire Category arguments, we…

We prove that if a Borel probability measure (\mu) on (\T) is invariant under the action of a "large" multiplicative semigroup (lower logarithmic density is positive) and the action of the whole semigroup is ergodic then (\mu) is either…

Consider a Moran-type iterated function system (IFS) \( \{\phi_{k,d}\}_{d\in D_{2p_k}, k\geq 1} \), where each contraction map is defined as \[ \phi_{k,d}(x) = (-1)^d b_k^{-1}(x + d), \] with integer sequences \( \{b_k\}_{k=1}^\infty \) and…

By seeing whether a Liouville type theorem holds for positive, bounded, and/or finite energy $p$-harmonic and $p$-quasiharmonic functions, we classify proper metric spaces equipped with a locally doubling measure supporting a local…

We study the eigenvalues and eigenfunctions of the Laplacian $\Delta_{\mu}=\frac{d}{d\mu}\frac{d}{dx}$ for a Borel probability measure $\mu$ on the interval $[0,1]$ by a technique that follows the treatment of the classical eigenvalue…

In this paper we investigate the following questions. Let $\mu, \nu$ be two regular Borel measures of finite total variation. When do we have a constant $C$ satisfying $$\int f d\nu \le C \int f d\mu$$ whenever $f$ is a continuous…

Recently, various extensions and variants of Bessel functions of several kinds have been presented. Among them, the $(p,q)$-confluent hypergeometric function $\Phi_{p,q}$ has been introduced and investigated. Here, we aim to introduce an…

The second part of the paper mainly deals with convergence of infinite determinantal measures, understood as the convergence of the approximating finite determinantal measures. In addition to the usual weak topology on the space of…

It is known that the tensor product of two sequences, in the tensor product of two separable Hilbert spaces, is a frame if and only if each component of that product is a frame. This paper proposes a sort of generalization of the…